The Bergman spaces of generalized analytic functions on the upper half-plane

For λ ≥ 0 λ ≥ 0 , a function f of $$z=x+iy$$ z = x + i y defined on a domain of the complex plane C C , symmetric about y -axis, is said to be λ λ -analytic if (Dₓ+i∂ y)f=0 ( D x + i ∂ y ) f = 0 , where Dₓ D x is the Dunkl operator on the real line given by Dₓφ (x)=∂ ₓφ (x)+(λ /x)( φ (x)-φ (-x)) D x φ ( x ) = ∂ x φ ( x ) + ( λ / x ) φ ( x ) - φ ( - x ) . In the paper we study the Bergman space Bλᵖ(C₊) B λ p ( C + ) associated with λ λ -analytic functions on the upper half-plane C₊ C + , and also its harmonic analog bλᵖ(C₊) b λ p ( C + ) . The reproducing kernels of Bλᵖ(C₊) B λ p ( C + ) and bλᵖ(C₊) b λ p ( C + ) are determined and the reproducing formulas of functions in Bλᵖ(C₊) B λ p ( C + ) and bλᵖ(C₊) b λ p ( C + ) are proved. The associated Bergman projections are proved to be bounded for 1<p<∞ 1 < p < ∞ , and the completeness for 1≤ p<∞ 1 ≤ p < ∞ and the duality for 1<p<∞ 1 < p < ∞ of Bλᵖ(C₊) B λ p ( C + ) and bλᵖ(C₊) b λ p ( C + ) are also considered.